QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  testmod Unicode version
Theorem testmod 1211
Description: A modular law experiment.
Assertion
Ref Expression
testmod (((c v a) v ((b v c) ^ (d v a))) ^ (a v (b ^ (d v ((a v c) ^ (b v d)))))) = ((b ^ ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d)))) v a)
Proof of Theorem testmod
StepHypRef Expression
1 leao1 162 . . . . . . . 8 ((a v c) ^ (b v d)) =< ((a v c) v ((b v c) ^ (d v a)))
21mli 1124 . . . . . . 7 ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d))) = (((a v c) v ((b v c) ^ (d v a))) ^ (d v ((a v c) ^ (b v d))))
3 orass 75 . . . . . . . 8 ((a v c) v ((b v c) ^ (d v a))) = (a v (c v ((b v c) ^ (d v a))))
43ran 78 . . . . . . 7 (((a v c) v ((b v c) ^ (d v a))) ^ (d v ((a v c) ^ (b v d)))) = ((a v (c v ((b v c) ^ (d v a)))) ^ (d v ((a v c) ^ (b v d))))
52, 4tr 62 . . . . . 6 ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d))) = ((a v (c v ((b v c) ^ (d v a)))) ^ (d v ((a v c) ^ (b v d))))
65lan 77 . . . . 5 (b ^ ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d)))) = (b ^ ((a v (c v ((b v c) ^ (d v a)))) ^ (d v ((a v c) ^ (b v d)))))
76ror 71 . . . 4 ((b ^ ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d)))) v a) = ((b ^ ((a v (c v ((b v c) ^ (d v a)))) ^ (d v ((a v c) ^ (b v d))))) v a)
8 an12 81 . . . . 5 (b ^ ((a v (c v ((b v c) ^ (d v a)))) ^ (d v ((a v c) ^ (b v d))))) = ((a v (c v ((b v c) ^ (d v a)))) ^ (b ^ (d v ((a v c) ^ (b v d)))))
98ror 71 . . . 4 ((b ^ ((a v (c v ((b v c) ^ (d v a)))) ^ (d v ((a v c) ^ (b v d))))) v a) = (((a v (c v ((b v c) ^ (d v a)))) ^ (b ^ (d v ((a v c) ^ (b v d))))) v a)
107, 9tr 62 . . 3 ((b ^ ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d)))) v a) = (((a v (c v ((b v c) ^ (d v a)))) ^ (b ^ (d v ((a v c) ^ (b v d))))) v a)
11 leo 158 . . . 4 a =< (a v (c v ((b v c) ^ (d v a))))
1211mli 1124 . . 3 (((a v (c v ((b v c) ^ (d v a)))) ^ (b ^ (d v ((a v c) ^ (b v d))))) v a) = ((a v (c v ((b v c) ^ (d v a)))) ^ ((b ^ (d v ((a v c) ^ (b v d)))) v a))
13 orcom 73 . . . . 5 (a v (c v ((b v c) ^ (d v a)))) = ((c v ((b v c) ^ (d v a))) v a)
14 or32 82 . . . . 5 ((c v ((b v c) ^ (d v a))) v a) = ((c v a) v ((b v c) ^ (d v a)))
1513, 14tr 62 . . . 4 (a v (c v ((b v c) ^ (d v a)))) = ((c v a) v ((b v c) ^ (d v a)))
16 orcom 73 . . . 4 ((b ^ (d v ((a v c) ^ (b v d)))) v a) = (a v (b ^ (d v ((a v c) ^ (b v d)))))
1715, 162an 79 . . 3 ((a v (c v ((b v c) ^ (d v a)))) ^ ((b ^ (d v ((a v c) ^ (b v d)))) v a)) = (((c v a) v ((b v c) ^ (d v a))) ^ (a v (b ^ (d v ((a v c) ^ (b v d))))))
1810, 12, 173tr 65 . 2 ((b ^ ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d)))) v a) = (((c v a) v ((b v c) ^ (d v a))) ^ (a v (b ^ (d v ((a v c) ^ (b v d))))))
1918cm 61 1 (((c v a) v ((b v c) ^ (d v a))) ^ (a v (b ^ (d v ((a v c) ^ (b v d)))))) = ((b ^ ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d)))) v a)
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1120
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by:  testmod1  1212
  Copyright terms: Public domain W3C validator